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Complex Analysis and Trig

  1. Jan 20, 2010 #1
    1. The problem statement, all variables and given/known data

    The principal valueof the logarithmic function of a complex variable is defined to ave its argument in the range -pi < arg(z) < pi. By writing z = tan(w) in terms of exponentials, show that:

    tan-1(z) = (1/2i)ln[(1 + iz)/(1 - iz)]


    3. The attempt at a solution

    I have absolutely no idea where to start on this problem. My brain must be fried this week, but all I know how to do is write z = tan(w) in terms of exponentials.

    z = (1/2i)(eiw + e-iw)/(eiw - eiw)


    Thanks!
    Andrew
     
  2. jcsd
  3. Jan 20, 2010 #2

    Dick

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    Homework Helper

    First you'd better check your expansion of z in terms of e^(iw) and e^(-iw). There's more than one mistake in there. Once you done that, put x=e^(iw). Then e^(-iw)=1/x. Solving for z in terms of x just means solving a quadratic equation.
     
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